∫e^x(sinx)^2dx
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∫e^x(sinx)^2dx
∫e^x(sinx)^2dx
∫e^x(sinx)^2dx
∫e^x(sinx)^2dx
=1/2∫e^x(1-cos2x)dx
=1/2∫(e^x-e^xcos2x)dx
=1/2∫e^xdx-1/2∫e^xcos2xdx
=1/2e^x-1/2∫e^xcos2xdx
∫e^xcos2xdx
=∫cos2xde^x
=e^xcos2x+2∫e^xsin2xdx
=e^xcos2x+2∫sin2xde^x
=e^xcos2x+2e^xsin2x-4∫e^xcos2xdx
5∫e^xcos2xdx=e^xcos2x+2e^xsin2x=e^x(cos2x+2sin2x)
∫e^xcos2xdx=1/5e^x(cos2x+2sin2x)
所以,原式1/2e^x-1/10e^x(cos2x+2sin2x)+C
∫e^x(sinx)^2dx
=∫(sinx)^2d(e^x)
=e^x(sinx)^2-2∫e^xsinxcosxdx
=e^x(sinx)^2-∫e^xsin2xdx
又
∫e^xsin2xdx
=∫sin2xd(e^x)
=e^xsin2x-2∫e^xcos2xdx
=e^xsin2x-2∫cos2xd(e^x)
=e^...
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∫e^x(sinx)^2dx
=∫(sinx)^2d(e^x)
=e^x(sinx)^2-2∫e^xsinxcosxdx
=e^x(sinx)^2-∫e^xsin2xdx
又
∫e^xsin2xdx
=∫sin2xd(e^x)
=e^xsin2x-2∫e^xcos2xdx
=e^xsin2x-2∫cos2xd(e^x)
=e^xsin2x-2e^xcos2x-4∫e^xsin2xdx
所以 5∫e^xsin2xdx=e^xsin2x-2e^xcos2x
∫e^xsin2xdx=(e^xsin2x-2e^xcos2x)/5
所以
原定积分=e^x(sinx)^2-(e^xsin2x-2e^xcos2x)/5+C
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